Used when people or things follow a sequence: taking turns, doing something in a fixed order, or handling items sequentially.
Please call the names in order.
Please enter the room one by one, in turn.
I arranged the books in order of size.
The teacher called out the names from the roll in order.
I answered the questions one by one, in order.
順に is a general term for doing things in order, while 順々に emphasizes a step-by-step, sequential progression through each item.
順次 is a formal, often written adverb for proceeding in order, while 順に is a neutral, everyday way to say the same thing.
順に means proceeding in the normal/set order, while 逆に means doing the same thing in reverse order.
From 順 (order, sequence) + adverbial particle に. The kanji 順 is a phono-semantic compound: 川 (river) + 頁 (head), originally meaning 'to follow the flow'.